Class Number Problem for Imaginary Cyclic Number Fields
نویسندگان
چکیده
منابع مشابه
The Gauss Class Number Problem for Imaginary Quadratic Fields
a result first proved by Heilbronn [H] in 1934. The Disquisitiones also contains tables of binary quadratic forms with small class numbers (actually tables of imaginary quadratic fields of small class number with even discriminant which is a much easier problem to deal with) and Gauss conjectured that his tables were complete. In modern parlance, we can rewrite Gauss’ tables (we are including b...
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Let N be an imaginary abelian number field. We know that hN , the relative class number of N , goes to infinity as fN , the conductor of N , approaches infinity, so that there are only finitely many imaginary abelian number fields with given relative class number. First of all, we have found all imaginary abelian number fields with relative class number one: there are exactly 302 such fields. I...
متن کاملThe Dirichlet Class Number Formula for Imaginary Quadratic Fields
because 2, 3, and 1± √ −5 are irreducible and nonassociate. These notes present a formula that in some sense measures the extent to which unique factorization fails in environments such as Z[ √ −5]. Algebra lets us define a group that measures the failure, geometry shows that the group is finite, and analysis yields the formula for its order. To move forward through the main storyline without b...
متن کاملThe Dirichlet Class Number Formula for Imaginary Quadratic Fields
Z[ √ −5] = {a+ b √ −5 : a, b ∈ Z}, because 2, 3, and 1± √ −5 are irreducible and nonassociate. These notes present a formula that in some sense measures the extent to which unique factorization fails in environments such as Z[ √ −5]. Algebra lets us define a group that measures the failure, geometry shows that the group is finite, and analysis yields the formula for its order. To move forward t...
متن کاملThe Dirichlet Class Number Formula for Imaginary Quadratic Fields
Z[ √ −5] = {a+ b √ −5 : a, b ∈ Z}, because 2, 3, and 1± √ −5 are irreducible and nonassociate. These notes present a formula that in some sense measures the extent to which unique factorization fails in environments such as Z[ √ −5]. The large-scale methodology deserves immediate note, before the reader is immersed in a long succession of smaller attention-filling specifics: • algebra lets us d...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 1998
ISSN: 0022-314X
DOI: 10.1006/jnth.1998.2298